c'r/.rfe© 

V^iQ-O 


Colorado 

State  Teachers  College 

BULLETIN 

SERIES  XX  JULY,  1920  NUMBER  4 


A,  Comparative  Study  of 
Three  Diagnostic  Arithmetic  Tests 

GEORGE  WILLIAM  FINLEY 


Published  Montlily  by  State  Teachers  College,  Greeley,  Colorado 
Entered  as  Second-Class  Matter  at  the  Posto-lfice  at  Greeley,  Colorado, 
under  the  Act  of  August  24,  1912 


A  COMPARATIVE  STUDY  . 
of  Three 

DIAGNOSTIC  ARITHMETIC  TESTS 

by 

George  William  Finley,  B.  S.,  M.  S., 
Professor  of  Mathematics  in  Teachers  College 
Greeley,  Colorado 


FOREWORD 


This  study  was  undertaken  with  a  view  to  making  a  comparison  of  the 
results  obtained  by  the  use  of  different  arithmetic  tests.  Those  chosen  for 
comparison  were  the  Cleveland  Survey  Tests,  the  Woody  Scale,  and  the 
Monroe  Diagnostic  Tests.  All  three  of  these  purport  to  be  diagnostic 
in  their  nature,  and  if  this  be  true  they  should  lead  to  approximately  the 
same  conclusions  concerning  the  arithmetical  abilities  of  the  children  tested. 
It  was  with  a  desire  to  determine  whether  they  do  this  or  not  that  this  study 
was  made. 

The  study  is  divided  into  two  parts.  Part  I  gives  a  discussion  of  the 
value  of  arithmetic  tests  in  general  and  a  description  of  the  tests  used. 
Part  II  gives  the  results  obtained  by  giving  the  three  different  tests  to  a 
group  of  children  and  the  conclusion  reached  from  these  results. 


A  Comparative  Study  of 
Three  Diagnostic  Arithmetic  Tests 


PART  I 

In  recent  years  there  has  been  a  most  remarkable  development  of  all 
kinds  of  educational  tests  and  measurements.  Of  course  it  has  always  been 
necessary  for  teachers  to  measure  their  pupils’  attainments  in  some  fashion 
or  other.  Some  children  were  promoted  at  the  end  of  the  year  while  others 
were  retained  in  the  same  grade.  This  was  done  because  the  teacher  judged 
that  in  the  one  case  sufficient  progress  had  been  made  to  enable  the  children 
to  do  the  work  of  the  next  grade,  while  in  the  other  such  progress  had  not 
been  made.  In  order  to  arrive  at  these  conclusions  the  teacher  had  to  measure 
the  achievements  of  the  various  children  in  the  grade.  Again  at  the  end  of 
each  month  teachers  were  called  upon  to  “grade”  the  pupils  in  the  various 
subjects  that  they  happened  to  be  studying.  This  again  called  for  the 
measuring  process.  But  the  sort  of  measuring  done  was  of  a  very  indefinite 
kind.  It  was  made  up  very  largely  of  the  teacher’s  estimates  of  the  child, 
and  into  it  entered  a  great  many  things  besides  the  ability  to  do  certain 
specific  things.  Then,  too,  the  teacher’s  knowledge  of  the  specific  abilities  of 
the  children  was  exceedingly  limited.  It  is  true  so-called  tests  and  examina¬ 
tions  were  given  but  they  were  of  such  a  nature  as  to  test  the  abilities  of 
the  children  only  in  a  very  general  way.  In  fact  they  were  often  said  to 
test  the  children’s  general  ability  in  this,  that,  or  the  other  subject,  whereas, 
as  Ave  now  know,  there  is  no  such  thing  as  general  ability  in  a  subject.  There 
are,  in  fact,  as  many  separate  abilities  in  even  a  single  subject  as  there  are 
different  types  of  mental  activities  involved. 

Another  difficulty  with  these  tests  was  that  they  lacked  uniformity.  If  a 
child  did  not  do  as  well  in  a  test  in  arithmetic  this  week  as  he  did  last  week 
it  was  taken  to  mean  that  he  was  losing  ground.  This  might  not  be  at  all 
true.  The  tests  were  different  and  therefore  there  was  really  no  basis  for 
comparison.  Again,  if  a  child  in  the  sixth  grade  got  a  grade  of  90%  in 
arithmetic  while  one  in  the  eighth  grade  got  a  grade  of  70'%  this  fact  did  not 
give  any  basis  for  comparing  the  abilities  of  these  two  children.  Their  grades 
were  obtained  upon  entirely  different  tests. 

This  then  was  the  state  of  things  up  to  within  the  last  twenty  years. 

At  the  present  time,  however,  quite  a  different  state  of  affairs  obtains. 
Tests  and  scales  have  been  developed  and  standardized  so  that  a  teacher 
need  no  longer  be  in  doubt  about  how  her  pupils  compare  with  other  pupils 
in  the  same  grade,  with  pupils  in  other  grades  of  the  same  school,  with 
pupils  in  other  school  systems,  or,  best  of  all,  with  their  own  jirevious  records 
in  any  specific  ability. 

The  Courtis  Standard  Research  tests  were  not  given  in  this  experiment, 
but  as  all  of  tlie  scales  have  been  built,  to  a  greater  or  less  extent,  upon  them 
they  will, be  discussed  here. 

Inspired  by  the  work  of  Rice  and  Stone,  the  pioneers  in  the  field  of  tests 
and  measurements  in  arithmetic,  Mr.  C.  A.  Courtis  took  up  the  task  of 
developing  a  set  of  standard  tests.  He  worked  out  a  set,  now  known  as  series 
A,  which  he  gave  to  thousands  of  children  in  different  parts  of  the  country. 
Five  thousand  children  were  tested  in  Detroit;  33,000  in  New  York;  20,000  in 
Boston,  and  many  others  in  smaller  systems.  In  scoring  these  ])apers  perhaps 
the  most  remarkable  fact  brought  out  was  the  wide  range  of  variability 
shown  by  the  children  in  any  given  grade.  Some  children  in  the  sixth  grade, 


for  instance,  made  scores  lower  than  the  average  of  the  third  grade  while 
others  exceeded  the  average  of  the  eighth  grade.  In  spite  of  this  fact,  however, 
Mr.  Courtis  found  that  the  scores  for  the  children  of  the  sixth  grade  tended 
to  be  grouped  about  a  certain  standard  of  excellence  which  was  a  little  lower 
than  that  about  which  the  scores  of  the  seventh  grade  children  tended  to  be 
grouped  and  higher  than  that  of  the  fifth  grade.  This  lead  to  the  establish¬ 
ment  of  certain  standards  of  excellence  for  the  different  grades  in  the  par¬ 
ticular  abilities  tested  by  these  examples. 

Series  A  of  the  Courtis  tests  includes  eight  separate  tests,  each  one  con¬ 
taining  more  examples  than  the  swiftest  child  could  complete  in  the  time 
allotted.  The  tests  are  thus  a  measure  of  speed  as  well  as  of  accuracy.  These 
eight  tests  take  up  the  combinations  in  addition,  subtraction,  multiplication 
and  division,  speed  copying  of  figures,  one-step  reasoning  problems,  abstract 
examples  in  the  four  fundamentals  and  two-step  reasoning  problems. 

After  using  this  series  for  several  years  Mr.  Courtis,  and  others  as  well, 
found  that  it  was  not  satisfactory  in  several  respects.  In  the  first  place  it 
was  too  expensive  in  both  thne  and  money.  Then  again  it  did  not  given  an 
adequate  test  of  the  abilities  most  needed  by  the  pupil.  It  tested  the  pupil’s 
knowledge  of  the  addition  combinations  but  did  not  give  much  information 
concerning  his  ability  to  apply  this  knowledge  to  the  addition  of  columns  of 
numbers.  The  same  is  true  of  the  other  operations.  He  found  also  that  there 
was  practically  no  relation  between  a  child’s  ability  to  give  the  addition 
combinations  and  his  ability  to  add  a  long  column  of  figures.  He  therefore 
devised  a  second  group  of  tests  known  as  series  B.  This  group  consists  of 
four  tests,  one  for  each  of  the  fundamental  operations. 

Test  1  involves  the  addition  of  columns  of  0  three-place  numbers;  Test 
2  the  subtraction  of  eight-place  from  eight-  and  nine-place  numbers;  Test  3 
the  multiplication  of  four-  by  tAvo-place  numbers,  and  Test  4  the  division  of 
four-  and  five-place  numbers  by  two-place  numbers.  These  tests  have  also 
been  thoroughly  standardized. 

These  Courtis  tests  are  of  great  value  to  the  teacher  or  supervisor  of 
arithmetic.  They  furnish  an  instrument  by  means  of  which  he  may  deter¬ 
mine  the  degree  of  excellence  reached  by  a  grade  or  an  individual  in  any 
one  of  the  four  fundamental  operations.  But  they  are  not  primarily  diagnostic 
in  their  nature.  Whatever  diagnosis  is  made  by  their  use  is  general  and  not 
specific  in  its  nature.  They  do  show,  for  instance,  that  a  certain  grade  is  low 
in  addition,  but  they  give  no  suggestion  as  to  just  which  one  of  the  several 
abilities  required  in  addition  is  at  fault.  Then,  too,  they  are  limited  to  the 
field  of  the  four  fundamental  operations  with  integers. 

Realizing  these  facts  a  number  of -investigators  have  been  at  work  devis¬ 
ing  tests  that  would  be  primarily  diagnostic  in  their  aim.  Three  such  tests 
or  scales  have  been  devised  and  used  to  a  considerable  extent,  viz.,  the 
Cleveland  Survej'^  tests,  the  Woody  scale,  and  the  Monroe  tests.  We  shall 
consider  them  in  the  order  given. 

THE  CLEVELAND  SURVEY  TESTS 

When  Dr.  Judd  and  his  co-laborers  started  the  Cleveland  Survey  they 
looked  over  the  field  of  existing  tests  and  scales  in  arithmetic  and  decided 
that  none  of  those  that  had  been  developed  up  to  that  time  would  meet  the 
needs  of  the  situation.  The  Courtis  tests  seemed  to  be  the  most  'promising 
but  they  were  open  to  serious  objections. 

Series  A  they  felt  to  be  unsatisfactory  for  the  same  reasons  as  those 
already  given  in  this  discussion.  Series  B  used  as  a  supplement  to  series  A 
would  constitute  a  decided  improvement.  But  even  this  combination  did  not 
go  far  enough  to  suit  them.  By  using  the  combination  they  saw  that  they 
could  measure  general  attainment  in  each  of  the  four  fundamental  operations 
but  nothing  more.  In  other  words  the  test  would  not  be  diagnostic.  For 


6 


instance,  a  pupil  might  shoM’  by  his  ^A'Ol•k  on  Test  1,  Series  A,  that  he  knew 
his  addition  tables  perfectly,  and  yet  he  might  fail  utterly  on  Test  1  of 
Series  B.  These  facts,  they  argued,  would  be  Avorth  knoAAung,  but  they  would 
be  of  comparatively  little  value  unless  supplemented  by  other  facts.  The 
question  of  why  he  failed  on  the  second  test  would  remain  unanswered.  It 
might  be  because  he  failed  “to  bridge  the  attention  spans,”  or  because  of  his 
inability  to  “carry,”  but  the  tests  would  give  no  indication,  as  to  which  it  was. 
In  order  to  throAV  light  upon  this  question  it  was  necessary  to  introduce 
betAveen  the  simple  types  of  the  first  series  and  the  more  complex  types  of 
the  second  some  intermediate  forms. 

These  investigators  accordingly  secured  the  co-operation  of  Mr.  Courtis 
and  worked  out  Avhat  are  knoAvn  as  the  Cleveland  Survey  Tests  in  Arithmetic. 
These  tests  are  here  reproduced  in  full.  .They  consist  of  15  sets,  designated 

A,  B, - 0.  There  are  four  sets  in  addition  (A,  E,  J,  M),  tAvo  in  subtraction 

(B,  F),  three  in  multiplication  (C,  G,  L),  four  in  division  (D,  I,  K,  N),  and 
tAvo  in  fractions  (H,  O).  This  gives  a  spiral  arrangement,  as  the  pupil  begins 
Avith  Set  A  and  takes  each  set  in  its  proper  order. 

In  the  sets  involving  addition.  Set  A,  which  is  simply  Test  1  of  Series  A 
in  the  Courtis  Standard  tests,  requires  simply  a  knowledge  of  the  combina¬ 
tions.  Set  E  requires  the  addition  of  columns  of  five  one-place  numbers.  This, 
then,  is  a  new  type.  The  pupil  must  combine  the  first  two  numbers  and  must 
then  hold  this  sum  in  mind  while  he  combines  it  in  turn  with  the  next  number. 
Set'  J  requires  the  addition  of  13  one-place  numbers.  This  again  introduces  a 
new  element,  “bridging  the  attention  span.”  It  is  a  Avell  known  fact  that  the 
addition  of  a  long  column  of  numbers  is  not  one  continuous  process.  The 
individual  rather  adds  up  several  numbers,  pauses  for  a  moment  Avhile  the 
attention  Avavers,  then  continues  the  addition.  The  fourth  set,  M,  requires 
the  addition  of  columns  of  five  four-place  numbers.  This  brings  in  another 
mental  process,  that  of  “carrying.”  The  four  sets  then  indicate  ability  or  lack 
of  ability  (1)  in  addition  combinations,  (2)  in  adding  several  numbers  in  a 
column,  (3)  in  “bridging  the  span  of  attention,”  and  (4)  in  “carrying.” 

The  tests  contain  but  two  sets  in  subtraction.  Set  B  tests  the  knowledge 
of  the  subtraction  combinations,  while  set  F,  the  subtraction  of  three-  from 
three-  and  four-place  iuimbers,  tests  a  knowledge  of  borrowing.  This  covers 
the  field  of  subtraction. 

In  multiplication  there  are  three  sets.  Set  C  gives  the  simple  combina¬ 
tions,  Set  G,  the  multiplication  of  four-place  by  one-place  numbers,  tests  a 
knowledge  of  “carrying,”  while  set  L,  ^le  multiplication  of  four-  by  two- 
place  numbers,  requires  a  knowledge  of  the  mechanics  of  handling  the  multi¬ 
plication  by  a  second  number  in  the  multiplier  and  of  the  addition  of  the 
partial  products. 

In  division  there  are  again  four  tests.  Set  D  tests  a  knowledge  of  the 
simple  combinations.  Set  I,  the  division  of  five-  by  one-place  numbers,  intro¬ 
duces  “carrying.”  Set  K,  the  division  of  three-  and  four-  by  tAvo-place  num¬ 
bers,  brings  in  the  simplest  type  of  long  division,  involving  no  carrying  in 
the  multiplication,  and  no  borrowing  in  the  subtraction.  Set  N  is  the  more 
complex  type  of  division  requiring  both  carrying  and  borrowing. 

These  tests  attempt  also  to  diagnose  the  ])upirs  ability  in  fractions  in 
addition  to  his  ability  in  the  fundamentals  with  integers.  For  this  purpose 
Sets  H  and  O  were  introduced.  Set  H  requires  addition  and  subtraction  of 
fractions  having  a  common  denominator,  Avdiile  in  Set  0  fractions  of  unlike 
denominators  are  added,  subtracted,  multiplied  and  divided. 

The  Cleveland  Survey  tests  carry  out  the  plan  of  the  Courtis  Standard 
tests  as  to  time  allowance.  The  time  limit  ranges  from  30  seconds  to  3  minutes, 
d'he  plan  Avas  to  give  sulficient  tinie  for  even  the  sloAvest  pupil  to  Avork  out 
at  least  one  example  but  not  enough  to  allow  the  swiftest  to  finish  them  all. 


7 


Aritl  metic  Exercises 
Cleveland  Survey  Tests 


Name . Age  today . 

Years  Months 

Grade .  School . .  Room . 

Teacher . . •  • . Date  today . 

Have  yon  ever  repeated  the  arittnnetic  of  a  grade  because  of  non-promo¬ 
tion  or  transfer  from  other  school?  If  so,  name  grade . 

Explain  cause . 


Inside  this  folder  are  examples  which  you  are  to  work  out  when  the 
teacher  tells  you  to  begin.  Woik  rapidly  and  accurately.  There  are  more 
problems  in  each  set  than  you  can  work  out  in  the  time  that  Avill  be  allowed. 
Answers  do  not  count  if  they  are  Avrong. 

Begin  and  stop  promptly  at  signals, from  the  teacher. 


1 

1 

1 

A  1 

1 

B  1  C 

1 

1 

1 

1 

1 

D  1  E 

1 

1 

1 

1 

F  1 

1 

G 

H 

A 

1 

1 

1 

1 

1 

1 

1 

1 

'l 

1 

1 

1 

i 

1 

1 

1 

1 

1 

1 

1 

R 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

Rank 

1 

1 

1 

1 

1 

i 

1 

1 

1 

1 

-V- 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1  I  1  J  1  K  1  L  1  :\I  1  N 

1  1  1  1  1  1 

1 

0 

1  1  1  1  1  1 

A  1  I  I  1  1  1 

1  1  1  1  1  1 

K  1  1  1  1  1  1 

1  1  1  1  1  1 

1 

1 

1  1  1  1  1  1 

Rank  III  II  1 

1  i  1  1  1  1 

8 


9 


SET  D — Division — 


3)9 

4)32 

6)36 

2)0 

6)48 

1)1 

5)10 

2)6 

8)32 

1)8 

5)30 

8)72 

2)10 

7)42 

1)1 

6)18 

1)3 

2)8 

6)6 

3)27 

5)0 

3)24 

9)63 

2)4 

6)42 

3)0 

7)21 

4)4 

SET  E- 

—Addition — 

5 

2  9 

2 

6 

2 

8  8 

8 

3 

2 

8  0 

5 

4 

0 

5  7 

0 

8 

4 

1  6 

6 

8 

6 

2  6 

8 

5 

7 

7  ■  2 

5 

9 

8 

3  3 

1 

6 

5 

4  9 

3 

3 

5 

1  3 

8 

.  8 

SET  F- 

—Subtraction 

616 

1248 

1365 

456 

.709 

618 

1267 

1335 

707 

509 

419 

277 

1355 

908 

519 

616 

258 

324 

1009 

768 

1269 

269 

295 

772 

SET  G- 

—Multiplication — 

2345 

9735 

8642 

2 

5 

9 

9735 

2468 

6789 

9 

3 

6 

5432 

9876 

8642 

4 

8 

5 

5432 

3689 

2457 

8 

5 

6 

Ats. 

Rts. 

7)28 

9)9 

3)21 

4)24 

7)63 

6)0 

1)0 

9)36 

1)7 

3)6 

4)20 

7)49 

8)64 

1)2 

4)16 

8)24 

7)7 

-2)  18- 

3)15 

9)81 

7)0 

1 

4 

9 

4 

6 

7 

2 

5 

1 

'5 

3 

5 

4 

4 

3 

4 

1 

3 

0 

4 

7 

8 

1 

2 

5 

8 

9 

5 

4 

6 

1092 

716 

472 

344 

816 

1157 

335 

908 

1236 

1344 

908 

818 

615 

854 

527 

286 

6789 

2345 

2 

6 

3579 

2468 

3 

7  • 

3579 

9876 

7 

4 

9863 

7542 

4 

. 

7 

10 


SET  H — Fractions- 


SET  H — Fractions — 

1 

Ats. 

Rts. 

3  1 

6  4 

4 

1 

8 

7 

- i - = 

- -  = 

— 

H - = 

— 

- = 

5  5 

9  9 

9 

9  • 

9 

9 

1  5 

3  1 

1 

4 

6 

2 

_ 1 - = 

- = 

— 

-1 - = 

— 

- = 

9  9 

7  7 

7 

7 

7 

7 

2  4 

4  1 

5 

1 

6 

5 

_ 1  _| _ = 

- - = 

— 

+  — = 

— •  - 

- = 

9  ■  9 

5  5 

8 

8 

9 

9 

7  1 

5  2 

5 

2 

8 

1 

- 1 - = 

- = 

H - = 

— 

- = 

.  9  9 

7  7 

9 

9 

9 

9 

1  3 

6  1 

2 

1 

5 

4 

_ _  _j _ = 

- — .  = 

■  +  — = 

— 

—  ■ — ■  = 

8  -  8 

8  8 

7 

7 

9 

9 

2  6 

4  3 

4 

2 

7 

5 

— +  — == 

- - = 

-  +  — = 

— 

- = 

9  9 

8  8 

7 

7 

9 

9 

SET  I — Division — 

4 ) 55424 

7)65982 

2)58748 

5)41780 

9)98604 

6)57432 

3)82689 

6)83194 

8)51496 

9)75933 

8)87856 

4)38968 

SP]T  J — iVddition — 

7  9  4 

7  2  9 

6 

7 

7  8 

9 

4 

3  2 

5  2  5 

1  9  6 

9 

1 

8  0 

5 

3 

1  1 

4  4  8- 

9  4  2 

6 

5 

5  7 

3 

7 

7  6 

2  8  1 

4  8  4 

7 

1 

4  1 

4 

7 

6  6 

0  7  8 

2  1  1 

4 

6 

8  5 

2 

2 

6  8 

6  2  4 

3  5  7 

0 

4 

1  8 

6 

0 

9  1 

5  5  5 

8  53 

3 

5 

2  1 

3 

9 

3  6 

1  3  1 

5  2  9 

7 

3 

1  3 

9 

5 

4  9 

8  6  3 

2  4  2 

1 

3 

3  7 

2 

6 

5  7 

3  1  9 

7  3  3 

6 

7 

9  4 

2 

3 

-4  5 

2  4  6 

7  6  8 

0 

6 

8  9 

8 

4 

2  2 

9  8  3 

1  7  5 

6 

1 

4  4 

5 

8 

9  2 

9  8  5 

9  6  5 

6 

7 

5  4 

6 

8 

9  4 

SET  K — Division — 

21)273  i 

52)1768 

41)779 

22)462 

31)837 

42)966  : 

23)483 

72)1656 

81)972  ■ 

73)1679 

21)294 

62)1984 

31)527 

52)2184 

41)984 

32)384  i 

51)2397 

82)1968 

71)3692 

22)484 

41)1681 

33)693 

61)1586 

53)1166 

31)496 

SET  L — Multiplication — 

8246 

3597 

5739 

2648 

29 

73 

85 

46 

4268 

7593 

6428 

8563 

37 

64 

58 

207 

— 

- - 

— 

— 

11 


SET  M— Addition- 


7493 

8937 

8625 

2123 

5142 

3691 

9016 

6345 

4091 

1679 

0376 

4526 

6487 

2783 

3844 

5555 

4955 

7479 

7591 

4883 

8697 

6331 

9314 

2087 

6166 

1341 

7314 

6808 

5507 

8165 

5226 

9149 

6268 

9397 

7337 

8243 

2883 

8467 

7725 

6158 

2674 

6429 

2584 

0251 

8331 

3732 

9669 

9298 

0058 

7535 

5493 

4641 

5114 

7404 

2398 

5223 

3918 

7919 

8154 

2575 

SET  N — Division— 

67)32763 

48)28464 

97)36084 

59)29382 

'78)69888 

88)34496 

69)40296 

38)26562 

SET  0 — Fractions- 

11  1 

9  1  ■ 

3  5 

- 1 - = 

— : - = 

—  X  —  = 

15  6 

14  4 

5  6 

5  2 

5  19 

11  5 

6  21 

_ _  _  _ 

6  20 

12  8 

1  3 

5  11 

5  2 

—  X  —  = 

- : - = 

- h  —  = 

6  10 

6  15 

12  8 

20  1 

3  3 

1  _ 

3  3 

21  6 

8  10 

Ats. 

Rts, 

Instructions  for  Examiners 


Have  the  children  fill  out  the  blanks  at  the  top  of  the  first  page.  Have 
them  start  and  stop  work  together.  Let  there  be  an  interval  of  half  a  minute 
between  the  sets  of  examples.  Take  two  days  for  the  test;  give  down  through 
I  the  first  day,  and  complete  the  test  on  the  next  day.  The  time  allowances 
below  must  be  followed  exactly. 


Set  A.  .  .  . 

Set  F.... 

.  .  1  minute 

Set 

K.  . 

Set  B...  . 

...  30  seconds 

Set  G. . . . 

.  .  1  minute 

Set 

L.. 

Set  C..  .  . 

.  .  .30  seconds 

Set  H.  .. 

...  30  seconds 

Set 

M.  . 

Set  D.  .  . 

...  30  seconds 

Set  I. . . . 

.  .  1  minute 

Set 

N.. 

Set  E . .  . 

...  30  seconds 

Set  J ... . 

...  2  minutes 

Set 

0.. 

2  minutes 

3  minutes 
3  minutes 
3  minutes 
3  minutes 


Have  the  children  exchange  papers.  Read  the  ansAvers  aloud  and  let  the 
children  mark  each  example  that  is  correct,  “C.”  For  each  set  let  them  count 
the  number  of  problems  attempted  and  the  number  of  “C’s”  and  write  the 
numbers  in  the  appropriate  columns  at  the  right  of  the  page. 

The  records  should  then  be  transcribed  to  the  first  page.  Please  verify  the 
results  set  down  by  the  pupils. 


12 


THE  WOODY  SCALES 


The  Woody  scales  are  the  results  of  another  attempt  to  devise  a  series 
of  tests  for  measuring  achievements  in  the  four  fundamental  operations  of 
aritlimetic.  The  author  of  the  scales  makes  the  statement  that  the  funda¬ 
mental  aim  was  to  devise  a  series  which  would  indicate  the  type  of  prob¬ 
lems  and  the  difficulty  of  the  problems  that  a  class  could  solve  correctly.  Each 
test  is,  therefore,  composed  of  as  great  a  variety  of  problems  as  possible.  They 
are  arranged  in  the  order  of  increasing  difficulty,  beginning  with  the  easiest 
that  can  be  found  and  gradually  increasing  in  difficulty  until  the  last  can  be 
solved  by  only  a  small  per  cent  of  the  pupil^  in  the  eighth. grade.  The  degree 
of  difficulty  of  each  problem  was  determined,  not  by  analysis,  but  by  sub¬ 
mitting  the  tests  to  a  large  number  of  children  and  computing  the  difficulty 
of  each  problem  from  the  number  of  children  that  were  able  to  solve  it. 

In  building  the  scales  under  the  above  outlined  plan  -the  author  made  up 
tests  containing  as  great  a  variety  of  problems  as  possible  and  submitted 
them  to  a  large  number  of  children.  The  results  of  these  tests  showed  that 
the  preliminary  tests  did  not  conform  to  the  plan  adopted.  They  did  not 
show  an  arrangement  of  problems  such  that  they  were  solved  by  a  gradually 
increasing  per  cent  of  the  pupils  from  one  grade  to  the  next  higher.  There 
were  large  gaps  between  certain  problems.  These  defects  were  remedied  by 
introducing  extra  problems  to  fill  up  these  gaps  and  by  dropping  out  such 
problems  as  were  solved  by  a  higher  percentage  of  pupils  in  the  lower  grades 
than  in  the  higher  grades. 

This  methbd  of  construction  has  been  severely  criticised.  It  is  main¬ 
tained  that  if  we  are  to  measure  arithmetical  abilities  with  any  degree  of 
certainty  we  must  include  in  our  tests  problems  that  exercise  all  the  im¬ 
portant  types  of  arithmetical  abilities,  whether  or  not  this  gives  us  a  list 
of  problems  gradually  increasing,  in  difficulty.  This  criticism  is  undoubtedly 
just  to  a  certain  extent.  At  least  it  is  safe  to  say  that  if  we  are  to  use  the 
Woody  scales  intelligently  we  must  know  their  limitations. 

These  scales  are  published  in  two  series,  A  and  B.  Series  A  is  the  more 
complete,  while  series  B  is  made  from  series  A  by  leaving  out  part  of  the 
problems,  and  is  intended  to  be  used  by  those  who  can  devote  but  a  limited 
time  to  giving  the  tests.  Series  A  was  used  in  this  study  and  is  given  here 
in  full. 


13 


Series  A 


Addition  Scale 
By  Clifford  Woody 

City .  County .  School .  Date .  .  . 

Name . .When  is  your  next  birthday? 

How  old  will  you  be? . Are  you  a  boy  or  a  girl?. .  .  . 

In  what  grade  are  you? . Teacher’s  name . 


(1) 

(2)  (3)  (4) 

(5) 

(6)  (7) 

(8) 

(9) 

2 

2  17  53 

72 

90  3  +  1  =• 

2  +  5  +  1 

=  20 

3 

4  2  45 

26 

37 

10 

0 

' 

Z 

30 

25 

(10) 

(11)  (12)  (13) 

(14)  (15) 

(16)  (17) 

(18) 

21 

32  43  23 

25  +,42  =  100 

9  199 

2563 

33 

59  1  25 

33 

24  194 

1387 

35 

17  2  16 

45 

12  295 

4954 

— 

—  13  — 

201 

15  156 

2065 

— 

46 

19  - 

— 

(19) 

(20)  (21) 

(22) 

(23)  (24) 

(25) 

$  .75 

$12.50  $8.00 

547 

]^  +  >^  =  4.0125 

%  +  %  +  78  +  Vs  = 

1.25 

16.75  5.75 

197 

1.5907 

.49 

15.75  2.33 

685 

4.10 

— 

-  4.16 

678 

8.673 

.94 

456 

6.32 

393 

— 

525 

240 

152 

(26) 

(27) 

(28)  (29)  (30) 

(31) 

(32) 

i2y. 

Vs  + 1/4  + 

%  +  1/4  =  43/4  2% 

113.46  3/^  +  %  +  1/4  = 

62% 

21/4  63/8 

49.6097 

12% 

5%  33/4 

19.9 

37% 

-  - 

9.87 

— 

.0086 

18.253 

6.04 

(3.3) 

.49 

(34) 

(35) 

(36) 

(37) 

.28 

/6  +  %  = 

2  ft.  6  in. 

2  yr,  5  mo. 

.63 

3  ft.  5  in. 

3  yr.  6  mo. 

121/8 

.95 

4  ft.  9  in. 

4  yr.  9  mo. 

21% 

1.69 

5  yr.  2  mo. 

323/4 

.22 

6  yr.  7  mo. 

— 

.33 

.36 

1.01 

.56 

•88  (38) 

.75  25.091  +  100.4  +  25  +  98.28  +  19.3614  = 

.56 
1.10 
.18 
.56 


14 


Series  A 

Subtraction  Scale 

By  Clifford  Woody 

City .  County .  School . .  Date. 

Name . When  is  yoiir  next  birthday? 

How  old  will  you  be  ? . Are  you  a  boy  or  girl  ? . 

In  what  grade  are  you  ? . •  •  . . Teacher’s  name . 


(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

(8) 

(9) 

(10)  (11) 

8 

6 

2 

9 

4 

11 

13 

59 

78 

7  —  4=  •  76 

5 

0 

1 

3 

4 

7 

8 

12 

37 

60 

(12) 

(13) 

(14) 

(15) 

(16) 

(17) 

(18) 

(19) 

(20) 

27 

16 

50 

21 

270 

393 

1000 

567482 

23/4  —  1  = 

3 

9 

25 

9 

190 

178 

537 

106493 

(21) 

(22) 

(23) 

(24) 

(25) 

(26) 

10.00 

372  — 

y2  = 

80836465 

27 

4  yds.  1  ft.  6  in. 

3.49 

49178036 

574 

1278 

2  yds.  2  ft.  3  in. 

(27) 

(28)  (29) 

(30) 

5  yds.  1  ft.  4  in. 

2  yds.  2  ft.  8  in. 

10  —  6.25=  ‘753/4 

5274 

9.8063  — 

9.019  = 

(31) 

(32) 

(33) 

(34)  (35) 

7.3  —  3.00081  = 

1912  6  mo.  8  da.  5 

2 

678  378  —  178  = 

1910  7  mo.  15  da. - =  278 


12  10 


15 


Series  A 


Multiplication  Scale 
By  Clifford  Woody 

City .  County .  School .  Date.. 

Xame . .* . When  is  your  next  birthday?. 

How  old  will  you  be ? . Are  you  a  boy  or  girl? . 


(1) 

(2) 

(.3) 

(4) 

0  llCVlllt 

(5) 

(6) 

(7) 

3X7  = 

5X1  = 

2X3  = 

4X8  = 

23 

310 

7X9  = 

3 

4 

(8) 

(9)  (10) 

(11) 

(12) 

(13) 

(14) 

(15) 

50 

254  623 

1036 

5096 

8754 

165 

235 

3 

6  "  7 

8 

6 

8 

40 

23 

(16) 

(17)  (IS) 

(19) 

(20) 

(21) 

(22) 

7898 

145  24 

9.6 

287 

24 

8X5%=  _ 

9 

206  234 

4 

.05 

21/0 

(23) 

(24) 

(25) 

(26) 

(27) 

(28) 

(29) 

11/4  X  8: 

=  16 

%  X  %  = 

9742 

6.25 

.0123 

1/8X2  = 

2% 

59 

3.2 

9.8 

(30) 

(31) 

(32) 

(33) 

(34) 

2.49 

12  15 

6  dollars  49 

cents 

21/2  X  3y2  = 

1/2  X  1/2  = 

.36 

25  32 

8 

(35) 

(36) 

(37: 

) 

(38) 

(39) 

987% 

3  ft.  5  ill. 

2%  X  41/2  X  11/2  = 

.09631/8 

8  ft.  91/2  in. 

25 

5 

.084 

9 

16 


Series  A 

Division  Scale 
By  Clifford  Woody 


City .  County.. 

Name.  .  .' . 

How  old  will  you  be  ? . 

In  what  grade  are  you  ? . 

.  School .  Date . 

. When  is  your  next  birthday  ? , 

(1) 

(2) 

(3)  (4) 

(5) 

(6) 

3)6 

9)27  4 

)28  1)5 

9)"^ 

3)39 

(7) 

(8)  (9) 

(10) 

(11) 

(1 

2) 

4^2  = 

9)0  ryr" 

6  X  .  .  .  .  =  30 

2)13 

2  - 

-  2  = 

(13) 

(14) 

(15)  (16) 

(!' 

7) 

4 )  24  lbs..  8  oz. 

8)5856  14  of  128  =  68)2108 

50  ^ 

-  7  = 

(18) 

(19) 

(20) 

(21) 

(22) 

13)65065 

248  --  7  = 

2,1)25.2  25)9750 

2)13.50 

(23) 

(24) 

(25) 

(26) 

-23)469 

75)2250300  2400)504000 

12)2.76 

(27) 

(28) 

(29) 

(30) 

Ys  of  624  = 

.003). 0936 

31/2  =  9  = 

3/4- 

-  5  = 

(31) 

5  3 

4  5 

(32) 

9%  --  33/4  = 

(33) 

52)3756 

(34) 

(35) 

(36) 

62.50  ^  11/4  = 

531)37722 

9)69  lbs.  9  oz. 

The  addition  scale  begins  with  2-1-3  and  includes  the  addition  of  in¬ 
creasingly  difficult  exercises.  It  brings  in  fractions,  both  with  common  denomi¬ 
nators  and  with  different  denominators,  mixed  numbers,  decimals,  and  com¬ 
pound  numbers  of  two  denominations. 

The  subtraction  scale  is  made  up  of  problems  involving  numbers  of  the 
same  kind  as  those  in  the  addition  scale. 

The  multiplication  scale  includes  the  simple  combinations,  multiplica¬ 
tion  of  integers  by  integers  up  to  four  figures  in  the  multiplicand  and  two 
in  the  multiplier,  a  fraction  by  a  fraction,  a  decimal  by  a  decimal,  and  a 
compound  number  by  an  integer. 

The  division  scale  includes  the  simple  combinations,  short  division,  long 
division  up  to  the  division  of  a  number  of  from  five  to  seven  digits  by  one 
of  two  or  three  digits,  division  of  mixed  numbers,  fractions,  decimals  and 
compound  numbers. 

fn  giving  these  tests  the  time  allowed  is  practically  unlimited;  twenty 
minutes  being  allowed  for  each  test.  In  this  length  of  time  nearly  all  the 
pupils  will  have  completed  all  the  problems  that  they  can  solve.  These  tests 

17 


are  then  “power”  tests  rather  than  “speed”  tests  such  as  those  devised  by 
Courtis. 

Another  way  in  which  the  Woody  scales  differ  from  the  Courtis  tests  is 
that  in  the  latter  the  problems  in  a  given  test  are  of  equal  difficulty, 
while  in  the  former  they  are  of  varying  degrees  of  difficulty.  This  being  the 
case  it  became  necessary  for  Mr.  Woody  to  adopt  some  unit  by  means  of 
which  the  degree  of  difficulty  of  each  problem  could  be  stated.  The  unit' 
adopted  was  the  Probable  Error  (P.  E.)  of  the  school  grade  distribution.  The 
median  achievement  of  a  grade  distribution,  i.  e.,  a  problem  that  is  solved 
by  exactly  50%  of  the  grade,  is  taken  as  the  measure  of  the  achievement  of 
the  grade.  The  P.  E.  of  a  grade  distribution  is  that  distance  along  the  base 
line  of  a  surface  distribution  from  the  median  point  to  the  perpendicular  on 
either  side  of  the  median  which  cuts  off  twenty-five  per  cent  of  the  cases. 
The  P.  E.  of  the  grade’s  distribution  is  the  limits  of  the  middle  50%  of 
the  grade.  In  other  words  if  exactly  50%  of  a  class  are  able  to  solve  a  problem 
correctly,  then  25%  of  that  class  should  be  -able  to  solve  a  problem  that  is 
at  least  one  unit  (P.  E.)  more  difficult,  and  75%  of  that  class  should  be 
able  to  solve  a  problem  one  unit  less  difficult. 

THE  MONROE  TESTS 

The  third  series  of  tests  included  in  this  study  is  the  one  devised  by 
Walter  S.  Monroe. 

This  author  starts  out  deliberately  to  construct  a  series  of  tests  of  the 
operations  in  arithmetic  that  will  include  all  or  nearly  all  of  the  types  of 
examples  encountered  in  arithmetical  work.  He  points  out  the  fact  that 
existing  studies  show  that  there  are  as  many  arithmetical  abilities  as  there 
are  types  of  examples  and  argues  that  any  test  that  is  to  be  really  diagnostic 
must  include  all  the  important  types  within  its  scope. 

According  to  Mr.  Courtis  there  are  six  types  of  operations  in  the  addition 
of  integers,  four  in  subtraction,  nine  in  multiplication,  and  ten  in  division. 
Kallom  has  analyzed  the  addition  of  two  fractions  and  reached  the  con¬ 
clusion  that  there  are  fourteen  types  of  examples.  Mr.  Monroe,  without  mak¬ 
ing  a  very  careful  analysis,  carries  the  discussion  of  types  on  through  frac¬ 
tions  and  decimals  and  reaches  the  conclusion  that  there  are  at  least  86 
significant  types  of  examples  in  the  fundamental  operations  of  arithmetic 
(integers,  30;  common  fractions,  36;  decimal  fractions,  20  to  40).  This  is 
exclusive  of  those  involved  in  the  writing  and  reading  of  numbers,  in  the 
tables  of  denominate  numbers,  and  in  the  solution  of  problems. 

The  21  tests  devised  by  Mr.  Monroe  contain  61  of  these  types.  These 
tests  are  given  in  limited  lengths  of  time  so  that  they  measure  both  speed 
and  accuracy.  In  this  respect  they  differ  from  the  Woody  tests  and  agree 
with  the  Cleveland  tests.  In  fact  Mr.  Monroe  argues  that  arithmetical  abilities 
are  “two  dimensional,”  and  that  any  attempt  to  measure  them  must  take  this 
fact  into  consideration.  He  admits,  however,  that  the  usual  class-room  pro¬ 
cedure  is  to  measure  power  only  Avithout  much  regard  to  speed. 

The  21  tests  are  given  here  in  full.  They  are  printed  in  four  different 
folders.  The  first  two,  containing  tests  1-11,  deal  Avith  the  four  fundamental 
operations  Avith  integers;  the  third,  tests  12-16,  deals  Avith  common  frac¬ 
tions;  the  fourth,  tests  17-21,  Avith  decimal  fractions.  The  fourth  folder  Avas 
not  used  in  this  study. 


18 


Part  I — Tests  1-6. 


Bureau  of  Educational  Measurements  and  Standards 

Kansas  State  Normal  School,  Emporia,  Kansas 

DIAGNOSTIC  TESTS  IN  ARITHMETIC 
Operations  With  Integers 
Devised  by  Walter  S.  Monroe 

Name . . Age  today . 

Years  Months 

City . •• .  Grade .  Room . 

School .  Teacher . Date  today . 

Instructions  to  Examiners 

Have  the  pupils  fill  out  the  blanks  at  the  top  of  this  page.  Have  them 
start  and  stop  work  together.  Use  a  stop  watch  if  one  is  available;  if  not, 
use  an  ordinary  watch  with  a  second  hand  and  exercise  care  to  allow  just  the 
exact  time  for  each  test.  Allow  an  interval  of  half  a  minute  or  more  between 
tests.  Require  the  pupils  to  close  the  folder  as  soon  as  the  signal  to  stop  is 
given,  in  order  to  make  certain  that  they  do  not  spend  this  rest  period  working 
on  the  next  test.  If  the  pupils  need  to  sharpen  pencils  before  going  on,  allow 
this  to  be  done.  The  following  time  allowances  must  be  followed  exactly: 

Test  1 — 30  seconds.  Test  4 — 1  minute. 

Test  2 — 30  seconds.  Test  5 — 3  minutes. 

Test  3 — 1  minute.  "  Test  6 — 2  minutes. 

Have  the  children  read  the  following  directions:  “Inside  this  folder  are 
examples  which  you  are  to  work  out  when  the  teacher  tells  you  to  begin.  Do 
not  open  this  folder  before  the  teacher  gives  the  signal.  Work  rapidly  and 
accurately.  There  are  more  examples  in  each  test  than  you  can  work  out  in 
the  time  that  will  be  allowed.  Answers  do  not  count  if  they  are  wrong.  Begin 
and  stop  promptly  at  signals  from  the  teacher.  Place  the  test  in  position  on 
your  desk  so  that  you  can  open  it  quickly  when  the  signal  is  given  to  begin, 
but  do  not  open  it  until  the  signal  is  given.” 

After  all  of  the  tests  have  been  completed  have  the  pupils  exchange  papers. 
Read  the  answers  aloud  and  have  the  children  mark  each  example  that  is 
correct  “C.”  Count  the  number  of  examples  attempted  and  the  number  of 
“C’s”  and  write  the  numbers  in  the  proper  spaces  at  the  top  of  the  tests. 
Examples  partially  completed  or  partially  right  are  not  counted. 

Before  collecting  the  papers  have  the  records  transcribed  to  the  first  page. 
The  teacher  should  verify  a  sufficient  number  of  records  to  make  certain  that 
the  pupils  hav«  marked  the  papers  and  transcribed  the  results  correctly. 


1  1  1 

Test  . 1  I  1  2  1  3 

1  1  1 

1 

4 

5 

6 

1  1  1 

Number  of  examples  attempted] . j . j . '. 

1 

1  1  1 

1  1'  1 

Number  of  examples  right . | . j . j . 

19 


Test  1— ADDITION. 


At 

Rt 


4  5 

7  0 

2  9 


2  0 

6  3 

7  8 


1  7  6  7  3  2 
1  2  8  7  8  4 
4  3  4  0  9  0 


3  9 

3  •  4 

G  5 


8  8  5  4  4  1  0  0  7  6  6  3 
0996  5  5211877 
521  1  8  77,43309 


At 


Test  2 

—SUBTRACTION. 

Rt.  .  .  . 

37 

94 

60 

27  39 

41 

77 

53 

5 

8 

3 

6  7 

8 

3 

—  • 

9 

65 

80 

•  92 

70  68 

58 

26 

43 

2 

4 

5 

3  2 

9 

9 

8 

95 

50 

36 

34  44 

25 

63 

57 

4 

7 

1 

8  6 

•  — 

7 

9  , 

At ... . 

Test  3- 

-MULTIPLICATION. 

Rt. . .  . 

6572 

6750 

5863 

3754 

2845 

6 

9 

2 

5 

8 

4936 

9327 

8274 

8409 

6391 

4 

7 

3 

6 

9 

5482 

8609 

3679 

2758 

4658 

2 

5 

8 

4 

7 

9653 

3174 

2874 

7901 

2179 

3 

6 

9 

2 

At ... . 

5 

Test  4- 

-DIVISION. 

Rt: . . . . 

8 ) 3840 

4)7432 

7)2534 

3)8430 

6)4680 

9)8577 

2)6370 

5)9310 

8)7512 

4)3820 

7 ) 9653 

3)5781 

6)6720 

9)5373 

2)5130 

20 


Test  5— ADDITION. 


7862 

6809 

8941 

5917 

5013 

7623 

7910 

4814 

1761 

5299 

9845 

9007 

5872 

6601 

8522 

6975 

3739 

3496 

1046 

1227 

— 

— 

— 

— 

8758 

2462 

1247 

4319 

2350 

9869 

3573 

2358 

3197 

4572 

1081 

5795 

2338 

6420 

7805 

4314 

5917 

6772 

9864 

1249 

— 

— 

- - - 

- - - 

Test 

6— DIVISION. 

82)3854 

43)1591 

63)3591 

94)4042 

83)5312 

42)672 

62)2108 

93)5022 

84)7140 

41)3567 

64)5312 

92)6624 

6772 

At.  .  .  . 

Rt.  .  .  . 

7864 

1249 

6028 

7883 

8975 

6535 

8240 

9005 

2340 

9869 

1573 

2319 

6794 

3203 

— 

— 

— 

6794 

3283 

7917 

5420 

7805 

4304 

4570 

7642 

9027 

8028 

7803 

9975  ' 

8758 

2462 

1247 

At ... . 

Rt 


74)2664 

31)1953 

21)1344 

53)4452 

71)5183 

32)2304 

23)782 

51 )2703 

73)6278 

33)1386 

24)984 

52)3484 

21 


Part  II — Tests  7-11. 


Bureau  of  Educational  Measurements  and  Standards 

Kansas  State  Normal  School,  Emporia,  Kansas 

DIAGNOSTIC  TESTS  IN  AKITHMETIC 
Operations  With  Integers 
Devised  by  Walter  S.  Monroe 

Name . Age  today . 

Years  Months 

City .  Grade .  Room . 

School .  Teacher . Date  today . 


Instructions  to  Examiners 

Have  the  pupils  fill  out  the  blanks  at  the  top  of  this  page.  Have  them 
start  and  stop  work  together.  Use  a  stop  watch  if  one  is  available;  if  not, 
use  an  ordinary  watch  with  a  second  hand  and  exercise  care  to  allow  just  the 
exact  time  for  each  test.  Allow  an  interval  of  half  a  minute  or  more  between 
tests.  Require  the  pupils  to  close  the  folder  as  soon  as  the  signal  to  stop  is 
given,  in  order  to  make  certain  that  they  do  not  spend  this  rest  period  working 
on  the  next  test.  If  the  pupils  need  to  sharpen  pencils  before  going  on,  allow 
this  to  be  done.  The  following  time  allowances  must  be  followed  exactly: 

Test  7 — 2  minutes.  Test  10 — 2  minutes. 

Test  8 — 3  minutes.  Test  11 — 4  minutes. 

Test  9 — 1  minute. 

Have  the  children  read  the  following  directions:  “Inside  this  folder  are 
examples"  which  you  are  to  work  out  when  the  teacher  tells  you  to  begin.  Do 
not  open  this  folder  before  the  teacher  gives  the  signal.  Work  rapidly  and 
accurately.  There  are  more  examples  in  each  test  than  you  can  work  out  in 
the  time  that  will  be  allowed.  Answers  do  not  count  if  they  are  wrong.  Begin 
and  stop  promptly  at  signals  from  the  teacher.  Place  the  test  in  position  on 
your  desk  so  that  you  can  open  it  quickly  when  the  signal  is  given  to  begin, 
but  do  not  open  it  until  the  signal  is  given."’ 

After  all  of  the  tests  have  been  completed  have  the  pupils  exchange  papers. 
Read  the  answers  aloud  and  have  the  children  mark  each  example  that  js 
correct  “C.”  Count  the  number  of  examples  attempted  and  the  number  of 
“C’s”  and  write  the  numbers  in  the  proper  spaces  at  the  top  of  the  tests. 
Examples  partially  completed  or  partially  right  are  not  counted. 

Before  collecting  the  papers  have  the  records  transcribed  to  the  first  page. 
The  teacher  should  verify  a  sufficient  number  of  records  to  make  certain  that 
the  pupils  have  marked  the  papers  and  transcribed  the  results  correctl3^ 


1  1  1 

Test  . 1  7  1  8 

1  1 

9 

10 

11 

1  1  1 
Number  of  examples  attempted.. | . | . 

1 

1  i 

1  1  ! 
Number  of  examples  right . | . | . 

1 

1 . 

22 


Test  7— ADDITION. 


7 

6 

6 

8 

2 

1 

2 

6 

8 

7 

7 

9 

3 

2 

6 

8 

0 

9 

9 

8 

5 

5 

9 

1 

3 

2 

3 

1 

0 

9 

3 

5 

6 

6 

7 

5 

5 

4 

8 

0 

1 

1 

1 

1 

0 

0 

4 

6 

7 

8 

8  ’ 

7 

7 

7 

1 

4 

7 

7 

5 

3 

5 

5 

0 

3 

7 

5 

4 

2 

4 

5 

3 

4 

6 

6 

4 

2 

4 

1 

5 

4 

5 

7 

5 

3 

2 

4 

6 

9 

7 

9 

7 

Test 

8- 

-MULTIPLICATION. 

4857 

5718 

36 

92 

9625 

6123 

23 

64 

1253 

5376 

38 

76 

Test  9— SUBTRACTION. 


739 

1852 

975 

367 

948 

906 

508 

1371 

1284 

447 

843 

966 

1910 

/  3o 

1056 

361 

478 

591 

831 

954 

1077 

360 

483 

704 

At . 

Rt . 

8  3  2  6  9  5  7 

9*9  4  3  7  8  8 

5  1  1  6  4  9  0 

7  8  4  4  9  7  2 

2  1  2  8.8  3  1 

0  7  6  9  3  3  8 

8  6  3  2  3  9  9 

4  5  3  0  9  0  6 

2  2  3  1  8  7  8 

1  1  1  2  0  7  '6 

9  1112  0  7 

6  0  2  2  2  3  1 

7  8  3  4  4  4  5 


At 


Rt 


6942 

4065 

58 

47 

7486 

9027 

75 

89 

3786 

5492 

49 

At 

53 

Rt 

1087 

516 

962 

821 

239 

•  325 

730 

1853 

897 

508 

162 

258 

877 

1190 

619 

618 

739 

257 

1328 

939 

1316 

872 

654 

827 

8 

3 

4 

8 

5 

2 

7 

4 

2 

9 

5 

5 

5 


23 


At.  . 

Test 

10— MULTIPLICATION. 

Rt.  . 

560 

807 

617  . 

840 

730 

609 

37 

59 

508 

80 

96 

70 

435 

790 

940 

307 

682 

870 

308 

60 

38 

42 

409 

40 

780 

502  ■ 

386 

150 

850 

401 

56 

68 

207 

90 

72 

80 

817 

460 

730 

605 

392 

590 

109 

30 

52 

84 

306 

30 

At.  . 

Test 

11— DIVISION. 

Rt.  . 

47)27589 

79)36893 

36)28296 

68)31824 

96)56064 

28)21980 

57)22572 

89)25365 

48)32304 

76)36708 

67)39932 

98)46844 

24 


Part  III.  Tests  12-16. 


Bureau  of  Educational  Measurements  and  Standards 

Kansas  State  Normal  School,  Emporia,  Kansas 

DIAGNOSTIC  TESTS  IN  ARITHMETIC 
Operations  With  Common  Fractions 

Devised  by  Walter  S.  Monroe 

Name . Age  today . 

Years  Months 

City .  Grade .  Room . 

School .  Teacher . Date  today . 


Instructions  to  Examiners 

Have  the  pupils  fill  out  the  blanks  at  the  top  of  this  page.  Have  them 
start  and  stop  work  together.  Use  a  stop  watch  if  one  is  available;  if  not, 
use  an  ordinary  watch  with  a  second  hand  and  exercise  care  to  allow  just  the 
exact  time  for  each  test.  Allow  an  interval  of  half  a  minute  or  more  between 
tests.  Require  the  pupils  to  close  the  folder  as  soon  as  the  signal  to  stop  is 
given,  in  order  to  make  certain  that  they  do  not  spend  this  rest  period  working 
on  the  next  test.  If  the  pupils  need  to  sharpen  pencils  before  going  on,  allow 
this  to  be  done.  The  following  time  allowances  must  be  followed  exactly: 

Test  12—1%  minutes 

Test  13 — 2  minutes. 

Test  14 — 1  minute. 

Have  the  children  read  the  follo\fing  directions:  “Inside  this  folder  are 
examples  which  you  are  to  work  out  when  the  teacher  tells  you  to  begin.  Do 
not  open  this  folder  before  the  teacher  gives  the  signal.  Work  rapidly  and 
accurately.  There  are  more  examples  in  each  test  than  you  can  work  out  in 
the  time  that  will  be  allowed.  Answers  do  not  count  if  they  are  wrong.  Begin 
and  stop  promptly  at  signals  from  the  teacher.  Place  the  test  in  position  on 
your  desk  so  that  you  can  open  it  quickly  when  the  signal  is  given  to  begin, 
but  do  not  open  it  until  the  signal  is  given." 

After  all  of  the  tests  have  been  completed  have  the  pupils  exchange  papers. 
Read  the  answers  aloud  and  have  the  children  mark  each  example  that  is 
correct  “C.”  Count  the  number  of  examples  attempted  and  the  number  of 
“C’s”  and,  write  the  numbers  in  the  proper  spaces  at  the  top  of  the  tests. 
Examples  partially  completed  or  partially  right  are  not  counted. 

Before  collecting  the  papers  have  the  records  transcribed  to  the  first  page. 
The  teacher  should  verify  a  sufficient’  number  of  records  to  make  certain  that 
the  pupils  have  marked  the  papers  and  transcribed  the  results  correctly. 


Test  15^ — 2  minutes. 
Test  16—2  minutes. 


1  1  1 

Test  . 1  12  13 

1  1  1 

14 

15 

16 

1  1 

Number  of  examples  attempted.. | . | . 

1 

1 . 

f  ( 

1 

1  1 

Number  of  examples  right . | . 1 . 

1 

1 . 

25 


Test  12.— ADDITION. 


At 


Reduce  your  answers  to  lowest  terms. 

Rt . 

I 

1 

3  2 

5  2 

— 

4 - = 

- 1 - = 

_ 1 - = 

6 

3 

10  5 

9  3 

5 

1 

1  1 

5  7' 

— 

H - = 

—  4--  = 

- 1 - = 

6 

2 

8  2 

■  6  12 

3 

1 

1  1 

1  7 

— 

H - = 

- 1 - = 

- 1 - = 

4 

2 

3  12 

2  10 

3 

5 

5  1 

1  5 

— 

+  — = 

- 1 - = 

- 1 - = 

4 

12  - 

8  4 

2  12 

I 

2 

4  7 

5  3 

— 

“1 - = 

- 1 - = 

- 1 - = 

6 

3 

5  10 

8  4 

Test  13— SUBTRACTION. 

Reduce  your  answers  to  lowest 

terms. 

At . 

Rt . 

3 

2 

5  3 

1  2 

— 

- = 

— - - 

- 

4 

5 

6-4 

2  7 

7 

1 

2  1 

5  2 

— 

- =— 

- - — - 

- - 

10 

6 

3  2- 

6  15 

3 

1 

7  1 

2  3 

— 

- 

- = 

1 - ^ 

4 

3 

9  6 

3  5 

5 

3 

3  2 

7  3 

— 

- = 

- 

- = 

6 

8 

4  7 

12  8 

5 

3 

8  4 

4  1 

— 

- = 

- - - 

- - - — -  -  =: 

6 

5 

15  9 

5  3 

Test 

14— MULTIPLICATION. 

At . : . 

Reduce  your  ansAvers 

to  lowest 

terms. 

Rt . 

2 

3  - 

2  3 

5  3 

— 

X  —  = 

—  X  —  = 

—  X  —  = 

3 

4 

5  7. 

12  5 

4 

2 

1  3 

1  1 

— 

X  —  = 

—  X  —  = 

—  X  —  = 

9 

5 

3  8 

2  3 

2 

3 

4  1 

7  4 

— 

X  —  = 

—  X  —  = 

—  X  —  = 

5 

4 

5  3 

12  7 

3 

1 

2  1 

1  1 

— 

X  —  = 

—  X  —  = 

—  X  —  = 

8 

4 

7  6 

3  2 

4 

5 

4  7 

1  •  3 

— 

X  —  = 

—  X  —  = 

—  X  —  = 

15 

8 

5  9 

6  10 

26 


Test  15— ADDITION 


Reduce  your  answers  to  lowest  terms. 


1  3  3 

-+-=  — + 

6  5  12 

4  1  1 

- 1 - =  — "  + 

9  6  3 

1  2  7 

2  3'  10 

3  5  1 

- 1 - =  -  - h 

8  6  7 

2  2  3 

- \ - =  - h 

5  3  10 


Test  16— DIVISION. 

Reduce  your  answers  to  lowest  terms. 


2  1  4 

5  3  7 

5  5  3 

6  8  7 

1  1  2 

2  3  3 

4  8  3 

7  11  5 

4  1  2 

5  2  5 


Rt 


3  1 

5  2 

4  5 

15  9 

1  3 

3  4 

1  1 

10  15 

4  3 

— +  — = 
7  5 


At 

Rt 


3  2 

8  3 

7  4 
12  9 

2  3 

3  4 

1  1 

4  6 

5  4 

12  9 


5 

8 

4 

7 

3 

8 

2 

5 

1 

4 

2 

3 

4 

5 

8 

9 

3 

4 

3 

7 


27 


Part  IV,  Tests  17-21. 


Bureau  of  Educational  Measurements  and  Standards 
Kansas  State  Xormal  School,  Emporia,  Kansas 

DIAGNOSTIC  TESTS  IN  ARITHMETIC 

^Multiplication  and  Division  of  Decimal  Fractions 
Devised  by  Walter  S.  Monroe 

Name . ^...Age  today . 

Years  Months 

City .  Grade .  Room . 

School .  Teacher . Date  today . 

Instructions  to  Examiners 

Have  the  pupils  fill  out  the  blanks  at  the  top  of  this  page.  Have  them 
start  and  stop  work  together.  Use  a  stop  watch  if  one  is  available;  if  not, 
use  an  ordinary  watch  with  a  second  hand  and  exercise  care  to  allow  just  the 
exact  time  for  each  test.  Allow  an  interval  of  half  a  minute  or  more  between 

tests.  Require  the  pupils  to  close  the  folder  as  soon  as  the  signal  to  stop  is 

given,  in  order  to  make  certain  that  they  do  not  spend  this  rest  period  working 
on  the  next  test.  If  the  pupils  need  to  sharpen  pencils  before  going  on,  allow 
this  to  be  done.  The  following  time  allowances  must  be  followed  exactly: 

Test  17 — 30  seconds.  Test  20 — 30  seconds. 

Test  18 — 30  seconds.  Test  21 — 30  seconds. 

Test  19 — 30  seconds. 

Have  the  children  read  the  following  directions:  “Inside  this  folder  are 
examples  which  you  are  to  work  out  when  the  teacher  tells  you  to  begin.  Do 
not  open  this  folder  before  the  teacher  gives  the  signal.  Work  rapidly  and 
accurately.  There  are  more  examples  in  each  test  than  you  can  work  out  in 
the  time  that  will  be  allowed.  Answers  do  not  count  if  they  are  wrong.  Begin 
and  stop  promptly  at  signals  from  the  teacher.  Place  the  test  in  position  on 
your  desk  so  that  you  can  open  it  quickly  when  the  signal  is  given  to  begin, 
but  do  not  open  it  until  the  signal  is  given.” 

After  all  of  the  tests  have  been  completed  have  the  pupils  exchange  papers. 
Read  the  answers  aloud  and  have  the  children  mark  each  example  that  is 
correct  “C.”  Count  the  number  of  examples  attempted  and  the  number  of 
“C’s”  and  write  the  numbers  in  the  proper  spaces  at  the  top  of  the  tests. 
Examples  partially  completed  or  partially  right  are  not  counted. 

Before  collecting  the  jDapers  have  the  records  transcribed  to  the  first  page. 
The  teacher  should  verify  a  sufficient  number  of  records  to  make  certain  that 
the  pupils  have  marked  the  papers  and  transcribed  the  results  correctly. 


Test  . 

1  1 

17  1  18  1 

1  1 

1  i 

19  1 

1  I 

1 

20 

1 

21 

Number  of  examples  attempted.  . 

1  1 
. 1 . 

1  1 

1 . 

1 

1 . 

1  1 

1 

'  I 

i 

Number  of  examples  I'ight 

1 

. 1  . 

-W 


28 


Test  17— DIVISION. 


At 


Rt 


The  correct  answer  for  each  example  with  the  exception  of  the  decimal 
point  is  given  at  the  side  immediately  after  the  letters  “Ans.”  Write  the 
answer  in  its  proper  position  and  place  the  decimal  point  in  its  proper  place. 
Place  ciphers  before  or  after  the  answer  when  they  are  necessary. 


.03)16.2 

Ans. : 

54 

.07)1.82 

.06)7.44 

Ans. : 

124 

.08). 952 

.02). 144 

Ans.: 

72 

.08)40.8 

.03)47.4 

Ans. : 

158 

.07)8.61 

.09)5.76 

Ans. : 

64 

.04). 348 

.02). 748 

Ans.: 

374 

.03)89.1 

.09)94.5 

Ans. : 

105 

.01)5.48 

.04)9.84 

Ans. : 

246 

.07)'.238 

Test  18— MULTIPLICATION. 
Place  the  decimal  point  correctly 


657.2 

67.50 

5.863 

.7 

.03 

.6 

46004 

20250 

35178 

932.7 

82 . 74 

8.409 

.08 

.4 

.07 

74616 

33096 

58863 

367.9 

27.58 

4 . 658 

.2 

.05 

.8 

7358 

13790 

37264 

574.6 

82.47 

7.462 

.06 

.9 

.02 

34476 

74223 

14924 

Test 

19— DIVISION. 

Ans. :  26 

.05).415 

Ans.:  83 

Ans.:  119 

.04)87.6 

Ans.:  219 

Ans.:  51 

.09)3.42 

Ans.:  38 

Ans.:  123 

.05). 965 

Ans.:  193 

Ans.:  87 

.06)51.0 

Ans.:  85 

Ans.:  297 

^  .05)6.85 

Ans.:  137 

Ans. :  548 

.06). 288 

Ans.:  48 

Ans.:  34  .08)44.8 

At . 

Rt . 

in  the  folloAving  products: 

Ans.:  56 

375.4 

28.45 

4.936 

.09 

.2 

.05 

33786 

5690 

24680 

‘  639.7 

54.82 

8.609 

.3 

.06 

.9 

i9191 

32892 

77481 

965.3 

31.74 

2.874 

.04 

.7 

.03 

38612 

22218 

8622 

834.7 

54.32 

7.842 

.5 

.08 

.4 

41735 

43456 

31368 

At 

Rt 


The  correct  answer  for  each  example  with  the  exception  of  the  decimal 
point  is  given  at  the  side  immediately  after  the  letters  “Ans.”  W.rite  the 
answer  in  its  })roper  position  and  place  the  decimal  point  in  its  proper  place. 
Place  ciphers  before  or  after  the  answer  when  they  are  necessary. 


.4)148. 

Ans. : 

37 

.9)65.7 

.7). 301 

Ans. : 

43 

.3)47.7 

.2).548- 

Ans. : 

274 

.4)744. 

.9). 756 

Ans. : 

74 

.8)672. 

5)  865 

Ans. : 

173 

.3)684. 

.2)7.92 

Ans. : 

396 

.4)352. 

.7)3.22 

Ans. : 

46 

.5)  .710 

.1)9.42 

Ans. : 

942 

.6). 852 

Ans. : 

73 

.6)1.68 

Ans. : 

28 

Ans. : 

159 

.6)8.34 

Ans.: 

139 

Ans. : 

186 

.3)117. 

Ans.: 

39 

Ans. : 

84 

.7)59.5 

Ans. : 

85 

Ans. : 

228 

.6)93.6 

Ans.: 

156 

Ans. : 

88 

.3)16.2 

Ans. : 

54 

Ans. : 

142 

.8)376. 

Ans. : 

47 

Ans. : 

142 

.2)74.2 

Ans. : 

371 

29 

Test  20— MULTIPLICATIOX. 


At 


Et.  . 

Place  the  decimal  point  correctly  in  the  following  products: 


487.5 

57.28 

6.294 

4065. 

967.5 

.62 

9.5 

.28 

5.1 

8.4 

302250 

544160 

176232 

207315 

712700 

61.32 

7.465 

7486. 

907.2 

14.53 

.17 

4.3 

.76 

.39 

6.2 

104244 

320995 

558936 

353808 

90086 

5.376 

8637. 

549.3 

84.74 

8.637 

.91 

2.4 

5.7 

.83 

1.6 

489216 

207588 

313101 

703342 

138192 

5194. 

784.1 

36.74 

2.893 

4936. 

.49 

.72 

3.5 

.68 

9.4 

254506 

564552 

128590 

196724 

463984 

Test  21— DIVISION. 


At 

Rt 


The  correct  answer  for  each  example,  with  the  exception  of  the  decimal 
point,  is  given  below  the  quotient,  after  the  letters,  “Ans.”  Write  the  answer 
in  its  proper  position  and  place  the  decimal  point  in  its  proper  place.  Place 
ciphers  before  or  after  the  answer  when  necessary. 


.47)2758.9 

8.2)38.54 

79.)  36.893 

•  .43)1591 

Ans. :  587 

Alls.*:*  47 

Ans.:  467 

Ans.:  37 

3.6)2829.6 

74.)  26.64 

.68)31.824 

3.1)1953. 

Ans.:  786 

Ans.:  36 

Ans.:  468 

Ans. :  63 

96.)  5606.4 

.63)35.91 

2.8)21.980 

94.)  4.042 

Ans. :  584 

Ans.:  57 

Ans.:  785 

Ans.:  43 

.57)22572. 

2.1)140.7 

89.)  253.65 

.53)4.452 

Ans.:  396 

Ans. :  67 

Ans. :  285 

Ans. :  84 

4'.8)  32304. 

83.)531.2 

.76)367.08 

4.2). 672 

Ans.:  673 

Ans.:  64 

Ans.:  483 

Ans.:  16 

30 


PART  II  ' 


Having  in  mind  the  purpose  and  character  of  the' tests  to  be  used  we  may 
now  turn  to  the  main  question  at  issue  in  tlie  study,  viz.,,  do  the  different 
tests  agree  as  to  results?  If  they  do  the  fact  may  be  taken  as  a  strong  indi¬ 
cation  that  they  are  all  well  suited  to  their  purpose.  If  they  disagree  then 
certainly  one  or  more  of  the  tests  is  faulty  in  some  respect  or  else  they  do 
not  measure  the  same  abilities. 

The  tests  were  given  on  six,  successive  school  days,  beginning  October  23, 

to  a  group  of  about  60  eighth  grade  pupils  in  Manhattan,  Kansas.  The  order 
followed  was  Cleveland  tests,  Monroe  tests,  and  Woody  scales. 

The  tests  were  all  given  and  the  scores  checked  by  the  author.  Care  was 
exercised  to  see  that  conditions  were  as  nearly  identical  in  the  different  tests 
as  it  was  possible  to  make  them. 

The  results  of  the  tests  are  shown  in  Tables  1  to  6,  and  diagrams  1  to  6. 

Table  I  shows  a  comparison  of  the  standard  scores  and  the  class  scores 
for  the  number  of  problems  solved  correctly  and  the  per  cent  of  accuracy  in 
each  of  the  Cleveland  tests.  The  standards  shown  here  are  the  averages  of 
the  Cleveland,  Grand  Rapids  and  St.  Louis  median  scores  in  the  8B  sections. 
Table  2  gives  the  standard  scores  and  class  scores  in  attempts  and  in  per  cent 
of  accuracy  for  the  Monroe  tests.  In  both  of  these  tables  the  tests  are  arranged 
in  such  order  as  to  bring  together  all  the  tests  in  each  of  the  four  fundamental 
operations.  Tables  3,  4,  5  and  6  show  the  results  of  the  Woody  tests. 

These  results  are  shown  in  graphic  form  in  diagrams  1,  2,  3  and  4.  In 
these  diagrams  the  horizontal  lines  represent  the  grades,  the  vertical  lines  the 
tests  and  the  figures  at  the  points  of  intersection  the  standard  scores  of 
the  different  grades  in  the  indicated  tests.  The  broken  line  represents  the 
class  scores  as  determined  by  this  series  of  tests. 


Comparison  of  Standard  and  Class  Scores 


Table 

1 — Cleveland  Survey  Tests 

Table 

2 — Monroe  Tests. 

Standard 

Class 

Standard 

Class 

Scores 

Scores 

Scores 

Scores 

Test 

Rts. 

Ac. 

Rts.  Ac 

Test 

Ats. 

Ac. 

Ats. 

Ac. 

A  . 

.  .29.8 

99 

24.5 

99 

1  . 

.  12.7 

100 

12.5 

100 

E  ..... 

.  7.8 

94 

5.2 

93 

7  . 

.  5.4 

79 

4.9 

81 

,T  . 

.  5.6 

78 

3.7 

70 

5  . 

.  6.1 

66 

•  5.4 

62 

M  . 

.  5.3 

76 

4.6 

87 

2  . 

.  8.9 

100 

7.9 

100 

B  . 

.  25.2 

99 

18.2 

95 

9  . 

.  8.5 

97 

8.1 

100 

F  . 

.  10.2 

90 

7.1 

83 

3  . 

.  6.2 

84 

5.6 

86 

C  . 

.  19.7 

89 

16.3 

87 

8  . 

.  6.5 

73 

6.1 

81 

G  .  .  . 

6  9 

88 

5.5 

90 

10  . 

.  6.6 

82 

4.9 

90 

L 

4  7 

69 

3.6 

69 

4  . 

.  4.6 

88 

4.9 

100 

D  . 

.  22.3 

97 

18.7 

98 

6  . 

.  4.5 

100 

3.4 

100 

I  . 

.  4.7 

84 

2.8 

70 

11  . 

.  3.4 

68 

3.0 

100 

K  . 

.  10.8 

95 

7.6 

94 

12  . 

.  9.8 

73 

7.8 

76 

N  . 

.  2.4 

81 

1.5 

68 

15  . 

.  8.5 

59 

6.5 

76 

II  . 

.  9.3 

77 

5.6 

89 

13  . 

.  7.8 

71 

6.8 

81 

0  . 

.  5.7 

68 

3.5 

47 

14  . 

.  13.5 

75 

9.6 

93 

16  . 

.  8.5 

59 

9.7 

82 

31 


Table 

3 — Woody  Addition  Scale 

Table  4 — Woody  Subtraction  Scale 

Xo.  of 

Xo.  Getting 

%  Getting 

Xo.  of 

Xo.  Getting 

%  Getting 

Problem 

EachProb. 

EachProb. 

Problem 

Each  Prob. 

Each  Prob. 

1 . 

98 

1 . 

. 58 

98 

o 

. ■  .  59 

100 

o 

. 59 

100 

3 . 

. 59 

100 

3. .  .' . 

. 58 

98 

4 . 

. 59 

100 

4 . 

. 59 

'  100 

5 . 

98 

5 . 

. 59 

100 

6.; . 

98 

6 . 

. 59 

100 

. 58 

98 

- 

100 

8 . 

. 58 

98 

8 . 

. 58 

98 

0 . 

. .58 

98 

9 

100 

10 . 

95 

10 . 

100 

11 . 

. 56 

95 

11 . 

. 57 

'96 

12 . 

. 55 

93 

12 . 

. 59 

100 

13 . 

100 

13 . 

. 58 

98 

14 . 

.  55 

93 

14 . 

. 57 

96 

15 . 

92 

15 . 

. .'.  55 

93 

16 . 

. 55 

93 

16 . 

. 57 

96 

17 . 

. 52 

88 

17 . 

90 

IS . 

. 55 

93 

18 . 

. 54 

92 

10 . 

92 

10 

50 

85 

20 . 

. 53 

90 

20 . 

. 53 

90 

21 . 

. 47 

SO 

21 . 

. 44 

75 

oo 

. 41 

70 

oo 

. 54 

92 

23 . 

92 

>3 

49 

84 

24 . 

. 49 

84 

24 . 

. 51 

87 

25 . 

90 

•^5 

43 

73 

26 . 

. 47 

SO 

26 . 

. 45 

76 

27 . 

87 

27 . 

. 37 

,  63 

28 . 

. 53 

90 

28 . 

. 45 

76 

20 . 

. 44 

75 

29 . 

. 51 

87 

30 . 

. 43 

73 

30 . 

. 45  . 

76 

31 . 

. 37 

63 

31 . 

. 39 

66 

32 . 

. 45 

76 

32 . 

. 31 

52 

33 . 

61 

33 . 

. 42 

71 

34 . 

. 48 

81 

34 . 

. 36 

61 

35 . 

. . . 36 

61 

35 . 

. 40 

68 

36 . 

. 36 

61 

37 . 

. 30 

51 

38 . 

. 24 

34 

Standard  Score.  9.01;  Class  Score,  S.76  Standard  Score,  7.G4;  Class  Score,  7.00 


32 


Table  5— 

-Woody  Multiplication  Scale 

Table  6 — Woody  Division  Scale 

No.  of 

No.  Getting 

%  Getting 

No.  of 

No.  Getting 

%  Gettin 

Problem 

EachProb. 

EachProb. 

Problem 

EachProb. 

Each  Prol 

1 . 

.  58 

98 

1 . 

.  .  .  .  55 

96 

2 . 

. 59 

100 

2 . 

.  .  .  .  57 

100 

3 . 

. 59 

100 

3 . 

. .  .  .  57 

100 

4 . 

.  59 

100- 

4 . 

.  .  .  .  57 

100 

5 . 

. 59 

100 

5 . 

. .  .  .  57 

100 

6 . 

.  59 

100 

6 . 

. ...  57 

100 

7 . 

. ^58 

98 

7 . 

. .  . .  57 

100 

8  . 

. 59 

100 

8 . 

.  .  .  .  55 

96 

9 

55 

93 

9 

55 

96 

10 . 

.  56 

95 

10 . 

. .  .  .  57 

100 

11 . 

. 58 

98 

11 . 

.  .  .  .  55 

96 

12 . 

. 58 

98 

12 . 

. . . .  55 

96 

13 . 

.  51 

86 

13 . 

.  .  .  .  51 

89 

14 . 

. 58 

98 

14 . 

.  .  .  .  57 

96 

15 . 

. . 56 

95 

15 . 

.  .  .  .  52 

91 

16 . 

. 44 

75 

16 . 

.  .  . .  51 

89 

17....... 

. 53 

90 

17 . 

.  .  .  .  52 

91 

18 . 

. 54 

92 

18 . 

.  .  .  .  41 

68 

19 . 

. 55 

93 

19 . 

. .  .  .  51 

89 

20 . 

. 55 

93 

20 . 

.  .  .  .  48 

84 

21 . 

. 56 

95 

21 . 

.  .  .  .  53 

93 

22 . 

. 56 

95 

22 . 

.  .  .  .  49 

86 

23 . 

. 54 

92 

‘23 . 

.  .  .  .  36 

63 

24 . 

.  53 

90 

24 . 

.  .  .  .  43 

75 

25 . 

. 48 

81 

25 . 

.  .  .  .  42 

74 

26 . 

. 45 

76 

26 . 

.  .  .  .  44 

77 

27 . 

.  53 

90 

27 . 

. . . .  49 

86 

28 . 

. 50 

85 

28 . 

.  .  . .  45 

79 

20 . 

. 54 

52 

29 . .  . 

60 

30 . 

. 51 

87 

30 . 

.  .  .  .  36 

63 

31 . 

. 52 

88 

31 . 

. .  .  .  39 

68 

32 . 

. 43 

73 

32 . 

.  . .  .  43 

75 

33 . 

. 46 

78 

33 . 

.  .  .  .  36 

63 

34 . 

.  42 

71 

34 . 

.  .  .  .  33 

58 

35 . 

. 34 

57 

35 . 

.  .  .  .  22 

39 

36 . 

. 34 

57 

36 . 

. .  .  .  8 

14 

37 . 

. 36 

61 

38 . 

. 25 

42 

39 . 

. 27 

46 

Standard  Score,  7.93;  Class  Score,  8.19 

Standard  Score 

,7.16;  Class 

Score,  7.1 

33 


As  a  whole  the  group  made  the  poorest  showing  in  the  Cleveland  tests 
and  the  best  in  the  Woody  tests.  This  is  undoubtedly  due  in  part  to  the  fact 
that  the  Cleveland  tests  were  given  first.  It  also  indicates  that  the  Cleveland 
standards  are  higher  than  either  of  the  others.  In  the  Cleveland  tests  the 
scores  are  all  below  standard;  only  one  of  them  reached  seventh  grade 
standard,  five  are  between  seventh  and  sixth,  seven  between  sixth  and  fifth, 
and  two  below  fifth  grade. 

In  the  Monroe  tests  the  score  in  one  test  is  above  standard,  those  in  six 
tests  are  between  seventh  and  eighth  grade  standards,  and  those  in  the 
three  remaining  tests  are  below  sixth  grade  standards.  On  the  Woody  scale 
three  are  above  standard  and  one  between  seventh  and  eighth  grades. 

There  is  then,  even  in  this  general  statement,  a  serious  discrepancy  be¬ 
tween  the  results  obtained  from  the  Woody  scales  and  those  obtained  from 
the  other  two  tests.  Using  the  first  named  the  teacher  or  supervisor  would 
be  led  to  the  conclusion  that  these  pupils  did  not  need  much  more  drill  on 
the  fundamentals.  Using  either  of  the  others  he  would  come  to  exactly  the 
opposite  conclusion. 

But  leaving  the  standards  out  of  consideration  let  us  see.  how  the  results 
agree  as  to  the  strength  or  weakness  of  the  group  tested  in  the  different 
operations.  Both  the  Cleveland  and  the  Monroe  tests  show  weakness  in 
addition,  the  former  to  a  greater  extent  than  the  latter,  a  lesser  degree  of 
weakness  in  subtraction  and  multiplication  and  irregularity  in  division  and  in 
fractions.  The  Woody  tests  agree  with  this  showing  in  a  general  way,  but 
they  put  subtraction  considerably  above  any  of  the  other  operations. 

Turning  now  to  a  study  of  the  particular  abilities  in  the  various  opera¬ 
tions  let  us  see  what  the  different  tests  show.  Test  A,  Cleveland,  shows  the 
group  to  be  below  sixth  grade  attainment  in  knowledge  of  addition  combina¬ 
tions.  The  Monroe  tests  do  not  include  problems  of  this  character,  but  the 
Woody  addition  scale  has  two  problems.  Nos.  1  and  7.  Neither  of  these  shows 
any  weakness  here  as  both  were  solved  correctly  by  all  but  one  member  of 
the  group. 

Test  E,  Cleveland,  addition  of  o  figure  columns  of  single  digits,  indicates 
slightly  better  than  fourth'  grade  attainment,  the  weakest  point  in  addition. 
Test  1,  Monroe,  3  figure  columns  of  single  digits,  shows  between  seventh  and 
eighth  grade  attainment,  the  highest  point  in  addition.  Of  course  these  tests 
are  not  identical  in  character  and  these  results  seem  to  indicate  that  they  are 
not  even  of  the  same  type.  Problem  2,  Woody  addition  scale,  a  column  of 
three  figures,  was  solved  correctly  by  everj’^  member  of  the  group,  showing 
no  weakness  in  this  character  of  work. 

Test  J,  Cleveland,  addition  of  long  columns  requiring  the  bridging  of  the 
memory  span,  shows  a  score  below  fifth  grade  attainment,  but  slightly  better 
than  test  E.  Test  7,  Monroe,  gives  a  score  below  seventh  grade  standard, 
the  weakest  point  in  addition.  The  Woody  scale  does  not  give  a  problem  of 
this  character. 

Test  M,  Cleveland,  column  addition  four  numbers  wide  and  five  deep, 
gives  a  score  equal  to  seventh  grade  standard,  the  highest  point  in  addition. 
Test  5,  Monroe,  of  exactly  the  same  character,  also  gives  a  score  equal  to 
seventh  grade  standard.  Problem  18,  Woody  addition  scale,  was  solved 
correctly  by  93%  of  the  class,  a  showing  which  agrees  fairly  well  with  the 
other  two  results. 

Test  B,  Cleveland,  subtraction  combinations,  shows  a  score  a  little  below 
sixth  grade.  The  Monroe  tests  do  not  include' this  type,  but  problems  1  to  7 
and  problem  '10,  Woody  subtraction  scale,  show  no  weakness  at  all,  being 
solved  correctly  by  practically  every  member  of  the  class. 

Test  F,  Cleveland,  subtraction  involving  borrowing,  gives  a  score  between 
fifth  and  sixth  grade  standards.  Test  9,  Monroe,  gives  a  score  between  seventh 
and  eighth  grade  standards.  Problems  16,  17,  18,  19  and  23,  Woody  sub- 


34 


DIAGRAM  1 


Median  Scores  in  Rights,  Cleveland  Test 


0rades 

62 

7^ 

62 

5^ 

4  f 

3/ 

Ad  Sub.  Mu/.  D/v.  Fr. 

7 

3 

4 

.3 

98  l 

s  A 

6  5. 

5  Z 

/. 

02  / 

97  6: 

9  4 

.7  2 

23  4 

7  /. 

08  2 

9. 

3  3 

70  7. 

f  s 

2  4. 

•<?  / 

9/  6 

./  4 

.0  <2 

0-^  4 

(  (i 

^  / 

4 

47  6 

.6  ^ 

S  U 

./  \/ 

9.0  7 

5  / 

6.d  3 

r  ^ 

.3  / 

^3 

'A 

5  5 

S  3 

8  3, 

r 

\ 

2  / 

6.6  6. 

4-  / 

5.4  4 

.9  2 

.<5-  / 

S 

5.S  2 

/ 

./  4 

.9  / 

N 

/ 

/ 

.9  3 

7/  S 

7  £ 

7  / 

4  7 

^ 

■/  / 

/7  / 

f  ^7 

& 

/  /. 

3  / 

2  7. 

± _ 

_ 2 

Z _ 0 

_ 2 

.7 

r<zsf^  AEJMBFCGLDf  H  N  H  O 


DIAGRAM  2 

Median  Scores  in  Attempts,  Monroe  Test 


35 


Grad<zs 
8  ^ 

7  <3 


6 


DIAGRAM  3 

Class  Scores,  Woody  Scales 


T<zsf3  Ad. 


Of  ^ 

^  7 

.6  4- 

7 

S3  - - 

7. 

3/ 

7. 

26 

6. 

95 

6. 

46 

S 

72 

5. 

99 

5. 

47 

5. 

53 

4 

// 

za 

4. 

OS 

3. 

99 

a. 

96 

/. 

83. 

Z 

JZ 

/. 

44 

16 

S9 

87 

9^ 

2/ 


Sub. 


Mul. 


Div. 


DIAGRAM  4 

Cleveland  and  Monroe  Medians 


36 


DIAGRAM  5 

Accuracy  Graphs,  Cleveland  Test 


DIAGRAM  6 


- 3fandard  Scores. 

- C/ass  Scores.- 


37 


traction  scale,  were  solved  correctly  by  96,  90,  92,  85  and  84  per  cent,  respec¬ 
tively,  or  an  average  of  89  per  cent,  which  indicates  a  weakness  comparable 
to  that  shown  by  the  Monroe  test. 

Test  C,  Cleveland,  multiplication  combination,  places  the  children  in  this 
group  below  sixth  grade  standard.  The  Monroe  tests  do  not  include  this 
type,  and  the  Woody  multiplication  scale  problems  1  to  4  and  7  again  fail 
to  show  any  weakness. 

Test  G,  Cleveland,  multiplication  of  numbers  of  4  digits  by  a  single 
digit,  rates  the  group  a  little  below  sixth  grade  standard.  Test  3,  Monroe, 
the  same  kind  of  exercises,  rates  them  a  little  below  seventh  grade,  while 
problems  8,  9,  10,  11,  12,  13  and  16,  Woody  multiplication  scale,  show  scores 
of  100,  93,  95,  98,  86  and  75  per  cents,  respectively.  Of  these  13  and  16 
show  decided  weakness.  These  require  multiplication  by  8  and  9.  Here  again, 
then,  the  three  tests  are  in  substantial  agreement. 

Test  L,  Cleveland,  multiplication  by  numbers  of  two  digits,  gives  the 
highest  score  made  in  the  Cleveland  multiplication  tests,  midway  between 
sixth  and  seventh  grades.  Test  8,  Monroe,  places  the  score  above  seventh 
grade  standard,  and  is  also  the  best  score  made  in  multiplication.  Problems 
15  and  26,  Woody  multiplication  scale,  give  scores  of  95  and  76  per  cents, 
respectively,  slightly  better  than  the  scores  for  tltc  preceding  type.  The  three 
tests  here  show  substantial  agreement. 

Test  D,  Cleveland,  division  combinations,  shows  the  class  to  be  below 
seventh  grade  standard,  while  problems  1  to  5,  7,  8,  9  and  12,  Woody  division 
scale,  give  scores  indicating  practically  no  weakness  at  all. 

Test  I,  Cleveland,  short  division,  gives  a  score  below  sixth  grade  standard, 
while  test  4,  Monroe,  gives  a  score  above  eighth  grade  standard.  This  is  a 
discrepancy  that  is  difficult  to  account  for.  It  would  seem  to  indicate  that 
the  small  amount  of  practice  the  children  received  in  short  division  in  taking 
the  Cleveland  tests  made  a  decided  difference  in  their  ability  to  perform  this 
process  when  they  took  the  second  test.  Problem  4,  Woody  division  scale, 
gives  a  score  of  96  per  cent.  Examination  of  the  papers,  however,  shows  that 
most  of  the  children  used  the  process  of  long  division,  so  that  the  result  gives 
no  information  concerning  their  ability  in  short  division. 

In  test  K,  Cleveland,  long  division  with  small  units  digits  in  the  divisor, 
the  children  scored  a  little  above  sixth  grade  standard.  In  test  6,  Monroe, 
they  scored  a  little  below  sixth  grade.  In  problems  23  and  33,  Woody  division 
scale,  they  scored  63%  and  63%  respectively,  a  close  agreement  throughout. 

Test  N,  Cleveland,  long  division,  where  the  units  digit  in  the  divisor  is 
large,  shows  sixth  grade  standard.  Test  II,  Monroe,  shows  a  score  a  little 
below  seventh  grade  standard.  Problem  16,  Woody  division  scale,  shows  a 
score  of  89%.  Here  again  we  have  substantial  agreement. 

The  tests  in  fractions  are  not  enough  alike  in  type  to  make  direct  com¬ 
parisons  of  value.  In  the  main,  however,  all  three  of  the  tests  show  the  class 
to  be  weak  in  their  knowledge  of  the  operations  with  fractions. 

There  is  lack  of  agreement  then  between  the  Woody  scale  and  the  other 
two  on  the  combinations  in  all  four  fundamental  operations,  column  addition 
of  three  single  digits,  and  short  division,  and  substantial  agreement  in  addi¬ 
tion  involving  carrying,  subtraction  involving  borrowing,  multiplication  by 
one  or  more  digits,  long  division,  and  fractions. 

Diagrams  5  and  6  show  a  comparison  of  the  accuracy  scores  as  obtained 
in  the  Cleveland  and  the  Monroe  tests  in  those  types  of  problems  that  occur 
in  both  sets.  Both  of  these  graphs  show  a  much  closer  approach  to  the 
standards  than  was  found  in  either  the  rights  for  the  Cleveland  test  or  the 
attempts  for  the  Monroe  tests.  They  both  indicate  that  the  children  do  not 
vary  H'om  the  standards  so  much  in  accuracy  as  they  do  in  speed.  In  the 
main  the  two  tests  show  rather  close  agreement  as  to  results,  the  excep- 


38 


tions  being  in  short  column  addition  involving  carrying,  where  the  Cleveland 
test  shows  the  higher  degree  of  accuracy,  short  division,  where  the  Monroe 
test  gives  the  better  showing,  and  long  division  with  large  units  digits  in  the 
divisors,  where  the  Monroe  test  again  gives  the  better  showing.  The  decided 
increase  in  accuracy  in  division  shown  by  the  Monroe  tests  over  the  Cleveland 
tests  is  probably  due  to  the  fact  that  by  the  time  the  children  came  to  the 
Monroe  tests  in  division  they  had  discovered  the  fact  that  the  division 
examples  all  come  out  without  a  remainder.  This  enabled  them  to  detect 
errors  and  correct  them. 

The  Woody  scales  do  not  give  any  adequate  measure  of  accuracy. 

This  study  shows  then  that  there  is  a  substantial  agreement  between 
the  results  obtained  by  using  the  Cleveland  tests  and  those  obtained  by  using 
the  Monroe  tests.  The  Monroe  standards,  however,  seem  to  be  distinctly 
lower  than  the  median  scores  obtained  by  the  use  of  the  Cleveland  tests  in 
Cleveland,  Grand  Rapids  and  St.  Louis.  Considering  the  fact  that  this  study 
was  made  in  October  while  the  Monroe  standards  are  mid-year  results  it 
would  seem  that  they  are  too  low. 

The  Woody  scale,  on  the  other  hand,  gives  results  that  differ  materially 
from  those  obtained  by  the  use  of  the  other  two  tests.  As  has  already  been 
noted  this  scale  places  the  class  above  standard  in  everything  but  addition 
and  not  far  below  standard  even  there,  while  both  the  others  show  them 
to  be  distinctly  below  standard  in  all  the  operations.  Then  it  fails  altogether 
to  show  weakness  in  the  combinations  and  in  the  simple  problems,  a  weakness 
clearly  shown  by  both  of  the  other  tests.  It  fails  also  to  show  differences 
in  the  abilities  in  these  simpler  operations  of  the  different  children  in  a 
class.  That  marked  differences  do  exist  was  clearly  shown  by  the  distribu¬ 
tions  on  the  score  sheets  for  both  the  other  tests.  The  reason  for  this  failure 
is  not  far  to  seek.  Even  if  a  child  does  not  know  his  combinations  he  can 
count  up  the  results  in  the  simpler  problems  and  thus  secure  correct  results 
if  he  has  plenty  of  time,  and  the  Woody  scales  give  practically  unlimited 
time,  for  most  of  the  children  finish  each  of  these  scales  in  less  than  the 
twenty  minutes  allowed. 

The  Woody  scale  would  seem  to  be  deficient  then  in  several  ways:  (1) 
a  test  in  fundamental  operations  should  measure  both  speed  and  accuracy, 
as  well  as  a  knowledge  of  the  process  involved,  (2)  the  number  of  problems 
of  each  type  is  too  few  to  give  an  adequate  measure  of  ability,  (3)  it  fails 
to  show  individual  differences  between  pupils  or  even  classes  in  all  of  the 
simpler  processes,  (4)  there  is  a  lack  of  definiteness  in  the  results  obtained 
for  the  particular  weakness  (for  instance,  the  results  of  the  tests  in  this 
study  show  that  the  class  is  below  standard  in  addition,  but  they  do  not  tell 
us,  except  in  a  very  indefinite  way,  how  they  compare  with  other  eighth 
grade  classes  in  column  addition,  in  bridging  the  memory  span,  etc.),  (5)  its 
results  are  of  little  value  in  measuring  individuals,  while  both  the  other 
tests  can  be  used  to  great  advantage  in  this  regard.  On  the  other  hand  the 
Woody  test  has  some  good  points.  It  covers  a  wider  field  than  either  of  the 
other  tests.  While  it  fails  on  the  combinations  and  simple  exercises,  at  least 
for  upper  grade  work,  it  does  show  strength  or  weakness  in  the  more  im¬ 
portant  exercises,  the  ones  that  are  most  needed.  It  is  in  fact  a  test  of  neither 
speed  nor  accuracy,  but  rather  a  test  of  power.  It  can  be  used  to  advantage 
to  determine  which  processes  have  been  mastered  by  a  class  and  which  ones 
are  still  beyond  them. 

The  Cleveland  tests  could  be  considerably  improved  by  putting  the  four 
fundamental  operations  in  fractions  into  four  different  tests  instead  of  running 
them  together  in  the  two  tests,  H  and  0.  The  arrangement  in  test  II  is  par¬ 
ticularly  bad.  In  all  of  the  tests  up  to  this  point  the  pupils  have  had  a  single 
operation  to  perform,  so  that  many  of  them  when  th^  come  to  test  H  and 
start  in  by  adding  the  first  two  fractions,  go  right  on  and  add  all  the  others. 
So  marked  is  this  tendency  that  the  results  obtained  from  this  test  as  it 
stands  are  practically  worthless. 


3‘.) 


0112 


05735283 


The  Monroe  tests  could  be  greatly  improved  by  printing  the  exercises 
in  multiplication  and  long  division  so  there  would  be  more  space  for  the  work. 
As  they  are  they  make  the  work  so  crowded  that  the  children  are  seriously 
hampered. 

I  This  study  shows,  then,  that  tests  of  the  Cleveland  Survey  type  are 
I  superior  for  the  purjDose  of  diagnosing  strength  or  weakness  in  the  opera- 
/  tions  of  arithmetic  and  that  those  of  the  Woody  type  are  decidedly  inferior 
in  this  .jegardL7  They  have  their  principal  value  in  determining  what  proc¬ 
esses  have  been  mastered  by  any  given  class.  Both  types  are  valuable,  but 
each  should  be  used  in  the  kind  of  diagnosis  for  which  it  is  best  fitted. 


